CPEG 852 Advanced Topics in Computing Systems The Dataflow Model of Computation
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1 CPEG 852 Advanced Topics in Computing Systems The Dataflow Model of Computation Dynamic Dataflow Stéphane Zuckerman Computer Architecture & Parallel Systems Laboratory Electrical & Computer Engineering Dept. University of Delaware 140 Evans Hall Newark,DE 19716, United States September 8, 2015 S.Zuckerman CPEG852 Fall 15 Dataflow MoC 1 / 42
2 Outline 1 What We Know of Dataflow So Far... Static Dataflow Examples Static Dataflow Features Static Dataflow Activity Templates Recursive Program Graphs Ordinary and Tail Recursions 2 Dynamic Dataflow Introduction Dynamic Dataflow Model of Computation Dynamic Dataflow I-Structures A Bit of Theory... Dynamic Dataflow Architectures Dynamic Dataflow Architectures 3 Homework S.Zuckerman CPEG852 Fall 15 Dataflow MoC 2 / 42
3 Outline 1 What We Know of Dataflow So Far... Static Dataflow Examples Static Dataflow Features Static Dataflow Activity Templates Recursive Program Graphs Ordinary and Tail Recursions 2 Dynamic Dataflow Introduction Dynamic Dataflow Model of Computation Dynamic Dataflow I-Structures A Bit of Theory... Dynamic Dataflow Architectures Dynamic Dataflow Architectures 3 Homework S.Zuckerman CPEG852 Fall 15 Dataflow MoC 3 / 42
4 Reminder: Dataflow Model of Computation S.Zuckerman CPEG852 Fall 15 Dataflow MoC 4 / 42
5 Reminder: Dataflow Model of Computation S.Zuckerman CPEG852 Fall 15 Dataflow MoC 4 / 42
6 Reminder: Dataflow Model of Computation S.Zuckerman CPEG852 Fall 15 Dataflow MoC 4 / 42
7 Reminder: Dataflow Model of Computation S.Zuckerman CPEG852 Fall 15 Dataflow MoC 4 / 42
8 Reminder: Dataflow Model of Computation S.Zuckerman CPEG852 Fall 15 Dataflow MoC 4 / 42
9 Conditional Expressions Figure : if/else construct S.Zuckerman CPEG852 Fall 15 Dataflow MoC 5 / 42
10 Conditional Expressions Figure : if/else schema S.Zuckerman CPEG852 Fall 15 Dataflow MoC 5 / 42
11 Conditional Expressions Figure : loop schema S.Zuckerman CPEG852 Fall 15 Dataflow MoC 5 / 42
12 Static Dataflow The Golden Rule Static Dataflow s Golden Rule... for any actor to be enabled, there must be no tokens on any of its output arcs... S.Zuckerman CPEG852 Fall 15 Dataflow MoC 6 / 42
13 Example Power Function long power ( i n t x, i n t n) { i n t y = 1; f o r ( i n t i = n; i > 0; --i) y *= x; r e t u r n y; } Figure : Computation: y = x n S.Zuckerman CPEG852 Fall 15 Dataflow MoC 7 / 42
14 Example Power Function Computing y = 2 n S.Zuckerman CPEG852 Fall 15 Dataflow MoC 8 / 42
15 Static Dataflow Features One-token-per-arc Deterministic merge Conditional/iteration construction Consecutive iterations of a loop can only be pipelined A dataflow graph activity templates Opcode of the represented instruction Operand slots for holding operand values Destination address fields Token value + destination S.Zuckerman CPEG852 Fall 15 Dataflow MoC 9 / 42
16 Static Dataflow Activity Templates S.Zuckerman CPEG852 Fall 15 Dataflow MoC 10 / 42
17 Recursive Program Graphs Extension over static dataflow concepts Graph must be acyclic (DAG, directed acyclic graph) One-token-per-arc-per-invocation Iteration is expressed in terms of tail recursion S.Zuckerman CPEG852 Fall 15 Dataflow MoC 11 / 42
18 Examples of Ordinary Recursion Factorial long factorial ( long n) { i f (n == 0) r e t u r n 1; e l s e r e t u r n n * fact (n -1); } S.Zuckerman CPEG852 Fall 15 Dataflow MoC 12 / 42
19 Example Factorial DFG for fact(3) S.Zuckerman CPEG852 Fall 15 Dataflow MoC 13 / 42
20 Example Factorial DFG for fact(3) S.Zuckerman CPEG852 Fall 15 Dataflow MoC 13 / 42
21 Example Factorial DFG for fact(3) S.Zuckerman CPEG852 Fall 15 Dataflow MoC 13 / 42
22 Example Factorial DFG for fact(3) S.Zuckerman CPEG852 Fall 15 Dataflow MoC 13 / 42
23 Example Factorial DFG for fact(3) S.Zuckerman CPEG852 Fall 15 Dataflow MoC 13 / 42
24 Example Factorial DFG for fact(3) S.Zuckerman CPEG852 Fall 15 Dataflow MoC 13 / 42
25 Example Factorial DFG for fact(3) S.Zuckerman CPEG852 Fall 15 Dataflow MoC 13 / 42
26 Example Factorial DFG for fact(3) S.Zuckerman CPEG852 Fall 15 Dataflow MoC 13 / 42
27 Example Factorial DFG for fact(3) S.Zuckerman CPEG852 Fall 15 Dataflow MoC 13 / 42
28 Factorial Tail Recursive version factorial(3) S.Zuckerman CPEG852 Fall 15 Dataflow MoC 14 / 42
29 Factorial Tail Recursive version factorial(3) S.Zuckerman CPEG852 Fall 15 Dataflow MoC 14 / 42
30 Factorial Tail Recursive version factorial(3) S.Zuckerman CPEG852 Fall 15 Dataflow MoC 14 / 42
31 Factorial Tail Recursive version factorial(3) S.Zuckerman CPEG852 Fall 15 Dataflow MoC 14 / 42
32 Factorial Tail Recursive version factorial(3) S.Zuckerman CPEG852 Fall 15 Dataflow MoC 14 / 42
33 Factorial Tail Recursive version factorial(3) S.Zuckerman CPEG852 Fall 15 Dataflow MoC 14 / 42
34 Factorial Tail Recursive version factorial(3) S.Zuckerman CPEG852 Fall 15 Dataflow MoC 14 / 42
35 Factorial Tail Recursive version factorial(3) S.Zuckerman CPEG852 Fall 15 Dataflow MoC 14 / 42
36 Factorial Tail Recursive version factorial(3) S.Zuckerman CPEG852 Fall 15 Dataflow MoC 14 / 42
37 Factorial Tail Recursive version factorial(3) S.Zuckerman CPEG852 Fall 15 Dataflow MoC 14 / 42
38 Factorial Tail Recursive version factorial(3) S.Zuckerman CPEG852 Fall 15 Dataflow MoC 14 / 42
39 Factorial Tail Recursive version factorial(3) S.Zuckerman CPEG852 Fall 15 Dataflow MoC 14 / 42
40 Factorial Tail Recursive version factorial(3) S.Zuckerman CPEG852 Fall 15 Dataflow MoC 14 / 42
41 Factorial Tail Recursive version factorial(3) S.Zuckerman CPEG852 Fall 15 Dataflow MoC 14 / 42
42 Factorial Tail Recursive version factorial(3) S.Zuckerman CPEG852 Fall 15 Dataflow MoC 14 / 42
43 Factorial Tail Recursive version factorial(3) S.Zuckerman CPEG852 Fall 15 Dataflow MoC 14 / 42
44 Factorial Tail Recursive version factorial(3) S.Zuckerman CPEG852 Fall 15 Dataflow MoC 14 / 42
45 Factorial Tail Recursive version factorial(3) S.Zuckerman CPEG852 Fall 15 Dataflow MoC 14 / 42
46 Factorial Tail Recursive version factorial(3) S.Zuckerman CPEG852 Fall 15 Dataflow MoC 14 / 42
47 Factorial Tail Recursive version factorial(3) S.Zuckerman CPEG852 Fall 15 Dataflow MoC 14 / 42
48 Factorial Tail Recursive version factorial(3) S.Zuckerman CPEG852 Fall 15 Dataflow MoC 14 / 42
49 Factorial Tail Recursive version factorial(3) S.Zuckerman CPEG852 Fall 15 Dataflow MoC 14 / 42
50 Factorial Tail Recursive version factorial(3) S.Zuckerman CPEG852 Fall 15 Dataflow MoC 14 / 42
51 Factorial Tail Recursive version factorial(3) S.Zuckerman CPEG852 Fall 15 Dataflow MoC 14 / 42
52 Factorial Tail Recursive version factorial(3) S.Zuckerman CPEG852 Fall 15 Dataflow MoC 14 / 42
53 Factorial Tail Recursive version factorial(3) S.Zuckerman CPEG852 Fall 15 Dataflow MoC 14 / 42
54 Factorial Tail Recursive version factorial(3) S.Zuckerman CPEG852 Fall 15 Dataflow MoC 14 / 42
55 Factorial Tail Recursive version factorial(3) S.Zuckerman CPEG852 Fall 15 Dataflow MoC 14 / 42
56 Factorial Tail Recursive version factorial(3) S.Zuckerman CPEG852 Fall 15 Dataflow MoC 14 / 42
57 Factorial Tail Recursive version factorial(3) S.Zuckerman CPEG852 Fall 15 Dataflow MoC 14 / 42
58 Factorial Tail Recursive version factorial(3) S.Zuckerman CPEG852 Fall 15 Dataflow MoC 14 / 42
59 Factorial Tail Recursive version factorial(3) S.Zuckerman CPEG852 Fall 15 Dataflow MoC 14 / 42
60 Factorial Tail Recursive version factorial(3) S.Zuckerman CPEG852 Fall 15 Dataflow MoC 14 / 42
61 Factorial Tail Recursive version factorial(3) S.Zuckerman CPEG852 Fall 15 Dataflow MoC 14 / 42
62 Factorial Tail Recursive version factorial(3) S.Zuckerman CPEG852 Fall 15 Dataflow MoC 14 / 42
63 Outline 1 What We Know of Dataflow So Far... Static Dataflow Examples Static Dataflow Features Static Dataflow Activity Templates Recursive Program Graphs Ordinary and Tail Recursions 2 Dynamic Dataflow Introduction Dynamic Dataflow Model of Computation Dynamic Dataflow I-Structures A Bit of Theory... Dynamic Dataflow Architectures Dynamic Dataflow Architectures 3 Homework S.Zuckerman CPEG852 Fall 15 Dataflow MoC 15 / 42
64 Introduction I Static Dataflow Static dataflow allows for loops to be created Simple rules: One-token-per-arc But static dataflow has many limitations: A static dataflow actor cannot fire until all its output arcs are empty. Cannot handle procedure calls Cannot handle tail recursions Cannot handle arbitrary recursions S.Zuckerman CPEG852 Fall 15 Dataflow MoC 16 / 42
65 Introduction II Recursive Program Graphs Recursive Program Graphs extend static dataflow relax its constraints Some features: One-token-per-arc-per-invocation (also known as one-token-per-link-in-lifetime) No deterministic merge needed Graph must be acyclic (DAG, directed acyclic graph) Procedure calls are allowed through the use of the apply special actor Tail-recursive calls are allowed In fact, iterative behaviors are expressed through tail recursion Recursion is expressed by runtime copying Use of tags No token matching is needed But: still not completely general-purpose: Cannot handle arbitrary recursions Cannot handle mutual function function calls (e.g., function g calls function f, and vice-versa) S.Zuckerman CPEG852 Fall 15 Dataflow MoC 17 / 42
66 Dynamic Dataflow Model of Computation Dynamic Dataflow solves the remaining problems recursive program graphs couldn t: There is no limitation on the number of tokens per arc anymore Tokens are tagged with a color This implies new firing rules we ll examine them later Resulted in the MIT Tagged Token dataflow computer S.Zuckerman CPEG852 Fall 15 Dataflow MoC 18 / 42
67 Dynamic Dataflow I Tagged Tokens Semantics Token = Tag + Value: [v, < u, s >, d] v Value u Activation instance s Destination actor d Operand slot Loops & Function Calls Should be executed in parallel as instances of the same graph Arc A container with different tokens that have different tags ( colors ) Stored in a token store A node can fire as soon as all tokens with identical tags ( colors ) are presented in its input arcs S.Zuckerman CPEG852 Fall 15 Dataflow MoC 19 / 42
68 Revisiting Ordinary Recursions Factorial S.Zuckerman CPEG852 Fall 15 Dataflow MoC 20 / 42
69 Revisiting Ordinary Recursions Factorial S.Zuckerman CPEG852 Fall 15 Dataflow MoC 20 / 42
70 Revisiting Ordinary Recursions Factorial S.Zuckerman CPEG852 Fall 15 Dataflow MoC 20 / 42
71 Revisiting Ordinary Recursions Factorial S.Zuckerman CPEG852 Fall 15 Dataflow MoC 20 / 42
72 Revisiting Ordinary Recursions Factorial S.Zuckerman CPEG852 Fall 15 Dataflow MoC 20 / 42
73 Revisiting Ordinary Recursions Factorial S.Zuckerman CPEG852 Fall 15 Dataflow MoC 20 / 42
74 Revisiting Ordinary Recursions Factorial S.Zuckerman CPEG852 Fall 15 Dataflow MoC 20 / 42
75 Revisiting Ordinary Recursions Factorial S.Zuckerman CPEG852 Fall 15 Dataflow MoC 20 / 42
76 Revisiting Ordinary Recursions Factorial S.Zuckerman CPEG852 Fall 15 Dataflow MoC 20 / 42
77 Revisiting Ordinary Recursions Factorial S.Zuckerman CPEG852 Fall 15 Dataflow MoC 20 / 42
78 Revisiting Ordinary Recursions Factorial S.Zuckerman CPEG852 Fall 15 Dataflow MoC 20 / 42
79 Revisiting Ordinary Recursions Factorial S.Zuckerman CPEG852 Fall 15 Dataflow MoC 20 / 42
80 Revisiting Ordinary Recursions Factorial S.Zuckerman CPEG852 Fall 15 Dataflow MoC 20 / 42
81 Revisiting Ordinary Recursions Factorial S.Zuckerman CPEG852 Fall 15 Dataflow MoC 20 / 42
82 Revisiting Ordinary Recursions Factorial S.Zuckerman CPEG852 Fall 15 Dataflow MoC 20 / 42
83 Revisiting Ordinary Recursions Factorial S.Zuckerman CPEG852 Fall 15 Dataflow MoC 20 / 42
84 Revisiting Ordinary Recursions Factorial S.Zuckerman CPEG852 Fall 15 Dataflow MoC 20 / 42
85 Revisiting Ordinary Recursions Factorial S.Zuckerman CPEG852 Fall 15 Dataflow MoC 20 / 42
86 Revisiting Ordinary Recursions Factorial S.Zuckerman CPEG852 Fall 15 Dataflow MoC 20 / 42
87 Revisiting Ordinary Recursions Factorial S.Zuckerman CPEG852 Fall 15 Dataflow MoC 20 / 42
88 Revisiting Ordinary Recursions Factorial S.Zuckerman CPEG852 Fall 15 Dataflow MoC 20 / 42
89 Revisiting Ordinary Recursions Factorial S.Zuckerman CPEG852 Fall 15 Dataflow MoC 20 / 42
90 Revisiting Ordinary Recursions Factorial S.Zuckerman CPEG852 Fall 15 Dataflow MoC 20 / 42
91 Revisiting Ordinary Recursions Factorial S.Zuckerman CPEG852 Fall 15 Dataflow MoC 20 / 42
92 Revisiting Ordinary Recursions Factorial S.Zuckerman CPEG852 Fall 15 Dataflow MoC 20 / 42
93 Revisiting Ordinary Recursions Factorial S.Zuckerman CPEG852 Fall 15 Dataflow MoC 20 / 42
94 Revisiting Ordinary Recursions Factorial S.Zuckerman CPEG852 Fall 15 Dataflow MoC 20 / 42
95 Revisiting Ordinary Recursions Factorial S.Zuckerman CPEG852 Fall 15 Dataflow MoC 20 / 42
96 Revisiting Ordinary Recursions Factorial S.Zuckerman CPEG852 Fall 15 Dataflow MoC 20 / 42
97 Revisiting Ordinary Recursions Factorial S.Zuckerman CPEG852 Fall 15 Dataflow MoC 20 / 42
98 Revisiting Ordinary Recursions Factorial S.Zuckerman CPEG852 Fall 15 Dataflow MoC 20 / 42
99 Revisiting Ordinary Recursions Factorial S.Zuckerman CPEG852 Fall 15 Dataflow MoC 20 / 42
100 Dynamic Dataflow The Apply Operator S.Zuckerman CPEG852 Fall 15 Dataflow MoC 21 / 42
101 Dynamic Dataflow The Apply Operator S.Zuckerman CPEG852 Fall 15 Dataflow MoC 21 / 42
102 Dynamic Dataflow The Apply Operator S.Zuckerman CPEG852 Fall 15 Dataflow MoC 21 / 42
103 Dynamic Dataflow The Apply Operator S.Zuckerman CPEG852 Fall 15 Dataflow MoC 21 / 42
104 Dynamic Dataflow The Apply Operator S.Zuckerman CPEG852 Fall 15 Dataflow MoC 21 / 42
105 Dynamic Dataflow The Apply Operator S.Zuckerman CPEG852 Fall 15 Dataflow MoC 21 / 42
106 Dynamic Dataflow The Apply Operator S.Zuckerman CPEG852 Fall 15 Dataflow MoC 21 / 42
107 Dynamic Dataflow Features Advantages: Most parallel model out there You can be as parallel as Dynamic Dataflow, but not more Can handle arbitrary recursion Shortcomings: Needs to implement a token tag matching unit Ideally: use of associative memory (i.e., use of key-value pairs)... But it is not cost effective Instead, hashing can be used. S.Zuckerman CPEG852 Fall 15 Dataflow MoC 22 / 42
108 Dynamic Dataflow I I-Structures Contrary to the von Neumann model, dataflow models try to avoid side effects. I-Structures were designed to allow memory exploitation by dataflow programs. Special memory which follows a single assignment rule Each access to the data structure consumes it entirely Basic concept of the I-Structure: Only consume the entry on a write An I-Structure is a data repository which obeys the single assignment rule An I-Structure entry is written once, but read many times A data structure has a nil (undefined) value on a write Once written, the data becomes immutable As long as the data has not be written to the structure, all reads are deferred S.Zuckerman CPEG852 Fall 15 Dataflow MoC 23 / 42
109 Dynamic Dataflow II I-Structures This allows a program to use a data structure before it is completely defined Data structures can be incrementally created or read Status Present: The element can be read but not written Absent: The element has been attempted to be read but the element has not been written yet (initial state) Waiting: At least one read request has been deferred Error: an element was written more than once, an or out-of-range memory access happened S.Zuckerman CPEG852 Fall 15 Dataflow MoC 24 / 42
110 Dynamic Dataflow III I-Structures Elementary Operations allocate: reserves space for the new I-Structure I fetch: get the value of the new I-Structure (deferred) I store: writes a value into the specified I-Structure element Node Construction Select think...= A[i] Assign think A[i] =... S.Zuckerman CPEG852 Fall 15 Dataflow MoC 25 / 42
111 Node Construction Select S.Zuckerman CPEG852 Fall 15 Dataflow MoC 26 / 42
112 Node Construction Assign S.Zuckerman CPEG852 Fall 15 Dataflow MoC 27 / 42
113 Determinacy, Determinism, and Indeterminacy An execution is deterministic when For a given set of inputs, the computation always produces the same set of outputs The resulting computation issues operations exactly in the same order each time it is executing. An execution is determinate when For a given set of inputs, the computation always produces the same set of outputs The order of individual (sets of) instructions does not matter for the end result An execution is indeterminate or non-determinate when There is no guarantee that a given set of inputs fed to the computation will always yield the same outputs if we fire the computation with the same inputs. S.Zuckerman CPEG852 Fall 15 Dataflow MoC 28 / 42
114 Dataflow Models General Features Not history-sensitive (i.e., doesn t depend on time) Clear & simple semantics which lead to determinacy Maximal parallelism can be reached It actually can be a problem... S.Zuckerman CPEG852 Fall 15 Dataflow MoC 29 / 42
115 A Word on Eager vs. Lazy Expression Evaluation Definition: Eager Evaluation A system of expression evaluation is said to be eager when all its input arguments are evaluated before they are being used. Definition: Lazy Evaluation A system of expression evaluation is said to be lazy when its input arguments are evaluated only at the time they are needed (i.e., when the system needs to read or maybe write a value). S.Zuckerman CPEG852 Fall 15 Dataflow MoC 30 / 42
116 A Word On Strictness I Intuitive Definition A mechanism is strict if it requires all its arguments to be evaluated from the innermost expression toward its outermost expression. As a result, if an inconsistent or error state is encountered somewhere during the reduction of an expression, it will propagate the error state to the outer expressions. Formal Definition A function f is strict if f you can read this as a function applied to an invalid expression results in (or is reduced to) an invalid result S.Zuckerman CPEG852 Fall 15 Dataflow MoC 31 / 42
117 A Word On Strictness II Definition: Non-Strict Expression Reduction A function f is non-strict if f v, v Non-strict functions evaluate expressions from the outermost level down to the innermost ones By doing so, they may end up simplifying expressions which could yield errors, but which will never be reached. This feature inherently allows for more loose expression evaluation, and, as a result, for more parallelism S.Zuckerman CPEG852 Fall 15 Dataflow MoC 32 / 42
118 Using a Strict Apply Actor Provides a lot of parallelism Proposes a call-by-value type of semantics But: we lose the substitution property (also called referential transparency ) In other words: we risk losing determinacy To avoid this, a strictness analysis must be performed (at compile time) S.Zuckerman CPEG852 Fall 15 Dataflow MoC 33 / 42
119 Referential Transparency I More Formal Definitions Wikipedia s An expression is said to be referentially transparent if it can be replaced with its value without changing the behavior of a program (in other words, yielding a program that has the same effects and output on the same input). The opposite term is referential opacity. Dr. Gao s... The only thing that matters about an expression is its value, and any subexpression can be replaced by any other equal in value Moreover, the value of an expression is, within certain limits, the same when ever it occurs... S.Zuckerman CPEG852 Fall 15 Dataflow MoC 34 / 42
120 Referential Transparency II More Formal Definitions Referential Transparency and the Substitution Property Let f, g be two procedures/functions such that f appears as an application inside of g; let g be the procedure obtained from g by substituting the text of f in place of the application; In languages which are referentially transparent, the specification of the function for g will not depend on any terminating property of f, or g=g S.Zuckerman CPEG852 Fall 15 Dataflow MoC 35 / 42
121 Strictness, Eagerness, Non-Strictness, and Laziness I There is very often a confusion between eager expression evaluation and strict expression reduction lazy expression evaluation and non-strict expression reduction Eager/lazy evaluation refers to operational behavior, i.e., how the program reads (evaluates) an expression. Strict/Non-strict refers to semantics, i.e., the mathematical meaning of a given expression Which is why say that we reduce expressions (which from a mathematical point of view, boils down to meaning evaluate ). In practice, non-strict and lazy can often be used interchangeably, but beware! There is a class of expression reduction schemes called lenient executions Lenient executions are non-strict S.Zuckerman CPEG852 Fall 15 Dataflow MoC 36 / 42
122 Strictness, Eagerness, Non-Strictness, and Laziness II Lenient executions are not eager : arguments are not necessarily all evaluated before evaluating an expression, Lenient executions are not lazy: arguments may be evaluated before they are needed S.Zuckerman CPEG852 Fall 15 Dataflow MoC 37 / 42
123 Dynamic Dataflow Architecture The MIT Tagged Token Dataflow Architecture Processor Wait-Match Unit: Tokens for unary operations: go straight through Tokens for binary operations: try to match incoming token and a waiting token with same instruction address and context id Success: both token proceed forward Failure: incoming token rightarrow waiting token memory, bubble (no-op) forwarded S.Zuckerman CPEG852 Fall 15 Dataflow MoC 38 / 42
124 Dynamic Dataflow Architecture The Monsoon Dataflow Processor S.Zuckerman CPEG852 Fall 15 Dataflow MoC 39 / 42
125 Outline 1 What We Know of Dataflow So Far... Static Dataflow Examples Static Dataflow Features Static Dataflow Activity Templates Recursive Program Graphs Ordinary and Tail Recursions 2 Dynamic Dataflow Introduction Dynamic Dataflow Model of Computation Dynamic Dataflow I-Structures A Bit of Theory... Dynamic Dataflow Architectures Dynamic Dataflow Architectures 3 Homework S.Zuckerman CPEG852 Fall 15 Dataflow MoC 40 / 42
126 Homework Definition: Fibonacci Sequence U 0 = 0 U 1 = 1, n N U n = U n 1 + U n 2 Let the definition of the Fibonacci function using a pseudo-c syntax be: unsigned long fib(unsigned long n); Write the pseudo-code for the recursive version of fib Write the pseudo-code for the tail recursive version of fib Write the pseudo-code for the iterative (loop) version of fib S.Zuckerman CPEG852 Fall 15 Dataflow MoC 41 / 42
127 Homework Definition: Fibonacci Sequence U 0 = 0 U 1 = 1, n N U n = U n 1 + U n 2 Let the definition of the Fibonacci function using a pseudo-c syntax be: unsigned long fib(unsigned long n); Write the pseudo-code for the recursive version of fib Write the pseudo-code for the tail recursive version of fib Write the pseudo-code for the iterative (loop) version of fib Push tokens! Select one of the three versions you wrote for fib, and draw its corresponding dataflow graph If it is the ordinary recursive version, use dynamic dataflow If it is the tail-recursive version, use recursive program graphs If it is the iterative version, use static dataflow S.Zuckerman CPEG852 Fall 15 Dataflow MoC 41 / 42
128 References I S.Zuckerman CPEG852 Fall 15 Dataflow MoC 42 / 42
Topic B (Cont d) Dataflow Model of Computation
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CPEG 852 Advanced Topics in Computing Systems The Dataflow Model of Computation
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